Problem source

For any random variables $X_1$ and $X_2$, state the relationship between $\E(aX_1+bX_2)$ and $\E(X_1)$ and $\E(X_2)$, where $a$ and $b$ are constants.
If $X_1$ and $X_2$ are independent, state the relationship between $\E(X_1X_2)$ and $\E(X_1)$ and $\E(X_2)$.
An industrial process produces rectangular plates. The length and the breadth of the plates are modelled by independent random variables  $X_1$ and $X_2$ with  non-zero means  $\mu_1$ and  $\mu_2$  and   non-zero standard deviations $\sigma_1$ and $\sigma_2$, respectively. Using the results in the paragraph above, and without quoting a formula for $\var(aX_1+bX_2)$, find the means and standard deviations of the perimeter $P$ and area $A$ of the plates. 
Show that $P$ and $A$ are not independent.
The random variable $Z$ is defined by  $Z=P-\alpha A$, where $\alpha $ is a constant. Show that $Z$ and $A$ are not independent if 
\[
\alpha \ne \dfrac{2(\mu_1^{\vphantom2} \sigma_2^2 +\mu_2^{\vphantom2}\sigma_1^2)}
{ \mu_1^2 \sigma_2^2 +\mu_2^2\sigma_1^2 + \sigma_1^2\sigma_2^2 }
\;.
\]
Given that $X_1$ and $X_2$ can each take values 1 and 3 only, and that they each take these values with probability $\frac 12$, show that $Z$ and $A$ are not independent for any value of $\alpha$.

Solution source

$\E(aX_1+bX_2) = a \E(X_1) + b\E(X_2)$ for any $X_1, X_2$
$\E(X_1X_2)=\E(X_1)\E(X_2)$. if $X_1, X_2$ are independent.

\begin{align*}
&& \E(P) &= \E(2(X_1+X_2)) = 2(\E[X_1]+\E[X_2]) \\
&&&= 2(\mu_1 + \mu_2) \\
&& \var(P) &= \E[\left ( 2(X_1+X_2) \right)^2] - \E[2(X_1+X_2)]^2 \\
&&&= 4\E[X_1^2+2X_1X_2+X_2^2] -4(\mu_1 + \mu_2)^2 \\
&&&= 4(\mu_1^2 + \sigma_1^2 + 2\mu_1\mu_2 + \mu_2^2 + \sigma_2^2) - 4(\mu_1 + \mu_2)^2  \\
&&&= 4(\sigma_1^2+\sigma_2^2) \\
&& \textrm{SD}(P) &= 2 \sqrt{\sigma_1^2+\sigma_2^2}\\
\\
&& \E(A) &= \E[X_1X_2]  = \E[X_1]\E[X_2] \\
&&&= \mu_1\mu_2 \\
&& \var(A) &= \E[(X_1X_2)^2] - (\mu_1\mu_2)^2 \\
&&&= (\mu_1^2+\sigma_1^2)(\mu_2^2+\sigma_2^2) -  (\mu_1\mu_2)^2\\
&&&= \mu_1^2 \sigma_2^2 + \mu_2^2 \sigma_1^2 + \sigma_1^2 \sigma_2^2\\
&& \textrm{SD}(A) &= \sqrt{\mu_1^2 \sigma_2^2 + \mu_2^2 \sigma_1^2 + \sigma_1^2 \sigma_2^2}
\end{align*}

\begin{align*}
\E[PA] &= \E[2(X_1+X_2)X_1X_2] \\
&= 2\E[X_1^2X_2] + 2\E[X_1X_2^2]\\
&= 2(\mu_1^2 + \sigma_1^2)\mu_2 + 2\mu_1 (\mu_2^2+\sigma_2^2)\\
&\neq 2(\mu_1 + \mu_2)\mu_1\mu_2 \\
&= \E[P]\E[A]
\end{align*}

\begin{align*}
&& \E[Z] &= \E[P] - \alpha \E[A] \\
&&&= 2(\mu_1+\mu_2) - \alpha \mu_1 \mu_2 \\
\\
&& \E[ZA] &= \E[PA - \alpha A^2] \\
&&&=  2(\mu_1^2 + \sigma_1^2)\mu_2 + 2\mu_1 (\mu_2^2+\sigma_2^2) - \alpha \E[A^2] \\
&&&= 2(\mu_1^2 + \sigma_1^2)\mu_2 + 2\mu_1 (\mu_2^2+\sigma_2^2) - \alpha \E[X_1^2]\E[X_2^2] \\
&&&= 2(\mu_1^2 + \sigma_1^2)\mu_2 + 2\mu_1 (\mu_2^2+\sigma_2^2) - \alpha (\mu_1^2+\sigma_1^2)(\mu_2^2+\sigma_2^2) \\
\text{if ind.} &&  \E[Z]\E[A] &= \E[ZA]\\
&& (2(\mu_1+\mu_2) - \alpha \mu_1 \mu_2) \mu_1\mu_2 &= 2(\mu_1^2 + \sigma_1^2)\mu_2 + 2\mu_1 (\mu_2^2+\sigma_2^2) - \alpha (\mu_1^2+\sigma_1^2)(\mu_2^2+\sigma_2^2)  \\
\Rightarrow && 2(\mu_1^2\mu_2+\mu_1\mu_2^2) - \alpha \mu_1^2\mu_2^2 &= 2(\mu_1^2\mu_2+\mu_1\mu_2^2) + 2\sigma_1^2\mu_2 + 2\sigma_2^2\mu_1 - \alpha (\mu_1^2+\sigma_1^2)(\mu_2^2+\sigma_2^2)  \\
\Rightarrow && \alpha ((\mu_1^2+\sigma_1^2)(\mu_2^2+\sigma_2^2)  - \mu_1^2\mu_2^2) &= 2(\sigma_1^2\mu_2 + \sigma_2^2\mu_1) \\
\Rightarrow && \alpha &= \frac{ 2(\sigma_1^2\mu_2 + \sigma_2^2\mu_1) }{\mu_1^2 \sigma_2^2 + \mu_2^2 \sigma_1^2 + \sigma_1^2 \sigma_2^2}
\end{align*}

Therefore if they are not independent if $\alpha \neq $ the expression.

\begin{array}{c|c|c|c|c|c}
& X_1 & X_2 & A & P & Z \\ \hline 
0.25 & 1 & 1 & 1 & 4 & 4-\alpha \\
0.25 & 1 & 3 & 3 & 8 & 8-3\alpha \\
0.25 & 3 & 1 & 3 & 8 & 8-3\alpha \\
0.25 & 3 & 3 & 9 & 12 & 12-9\alpha \\
\end{array}

If $\mathbb{P}(A = 1, Z = 4-\alpha) = \mathbb{P}(A = 1)\mathbb{P}(Z = 4-\alpha)$ then $\mathbb{P}(Z = 4-\alpha) = 1$, but that mean $4-\alpha = 8-3\alpha = 12-9\alpha$ which is not a consistent set of equations as the first two are solved by $\alpha = 2$ and the second by $\alpha = \frac23$